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Journal of Integer Sequences, Vol. 15 (2012),
Article 12.7.8
23 11
Fibonacci Numbers of
Generalized Zykov Sums
César Bautista-Ramos and Carlos Guillén-Galván
Facultad de Ciencias de la Computación
Benemérita Universidad Autónoma de Puebla
14 Sur y Av. San Claudio, Edif. 104C, 303
Puebla, Pue. 72570
Mexico
bautista@cs.buap.mx
cguillen@cs.buap.mx
Abstract
We show that counting independent sets in several families of graphs can be done
within the framework of generalized Zykov sums by using the transfer matrix method.
Then we calculate the generating function of the number of independent sets for families
of generalized Zykov sums. We include many interesting particular cases (Petersen
graphs, generalized Möbius ladders, carbon nanotube graphs, among others).
1
Introduction
The Fibonacci number F (G) of a graph G was introduced by Prodinger and Tichy [15] and
is defined as the number of independent sets of G. In combinatorial chemistry this number is
also known as the Merrifield-Simmons index [11, 12]. Prodinger and Tichy calculated F (G)
recursively by paying attention to whether certain vertices appear or not in what they call
the usual recursion argument. Such binary occurrence problem is formalized, in this paper,
in the transfer matrix method [2, 6, 7, 10].
Our goal is to show how to count independent sets in graphs with some pattern structure.
The structure we are dealing with is a generalization of the Zykov sum of graphs (also known
as the graph join). Let us recall that the Zykov sum is the graph obtained from the disjoint
union of two graphs G1 , G2 by joining each vertex of G1 with each vertex of G2 . This join
is just a particular case of a relation set between the vertices of G1 and G2 . In this work,
1
we generalize Zykov sums by joining vertices only when they belong to a given relation set.
We call these open or closed Zykov sums (the formal definitions are introduced in the next
section). It is almost trivial to show that the independent sets of G1 and G2 give rise to a new
independent set in their open or closed Zykov sum, unless their vertices belong to the relation
(see Theorem 5). Such binary information, whether vertices are related or not, is stored in a
matrix known as the transfer matrix [2, 6, 7, 10]. The transfer matrix is particularly useful
for graphs that can be written as a repetitive pattern of open or closed Zykov sums. Then,
the product of the transfer matrices stores information about the number of independent
sets. We show that the usual matrix product gives the number of independent set in graphs
that are some kind of strip, while the Hadamard product of matrices is for closing such strips
in order to form some type of circular structure, like cycles or tori.
To illustrate our methods we use several families of graphs having a repetitive pattern of
generalized Zykov sums which include paths, cycles, grids, cylinders, tori, Möbius ladders,
and generalized Petersen graphs, among others. Additional examples are honeycomb tubes
(nanotube graphs) and honeycomb torus graphs (nanotorus graphs) which are particularly
interesting because they appear in parallel computing architectures [18] and nanotechnology
[14].
There are other works dealing with the Fibonacci number F (G) when G has a different
structure from those studied here; for instance, trees [11], regular graphs [3, 17, 21], unicyclic
graphs [13], graphs with small maximum degree [9], graphs with a given number of vertices
and edges [4], and graphs with a given minimum degree [8]. However, these works deal with
estimations for F (G), in contrast to our work, which is interested mainly in exact formulas
for F (G) when G is an open or closed Zykov sum. Exact formulas for Fibonacci numbers of
graph are also the concern of Golin, Leung, Wang, and Yong [10], Euler [7], Prodinger and
Tichy [15], Warner [20], Engel [6], Calkin and Wilf [2], Burnstein, Kitaev, and Mansour [1],
among others. However, there are some errors in [1], as we show in Section 5.
This paper is organized as follows: In Section 2 we introduce the basic definitions and
some examples; in Section 3 we give the related theorems for counting independent sets
along with some examples; in Section 4, we calculate the generating function of the Fibonacci numbers for some families of open and closed Zykov sums. Finally, in Section 5, we
describe more examples, among them (almost) regular graphs [1] where we also show some
counterexamples to the calculations in [1].
2
Definitions
In this paper we deal only with multigraphs [5]. However, since we are interested in the number of independent sets, the multiple edges are irrelevant, but the loops are not. Therefore,
by a graph, we refer to a multigraph without multiple edges but, perhaps, with loops. For
such a graph G, we denote the sets of vertices and edges of G as V (G) and E(G), respectively. We are mainly interested in binary relations between graphs because they describe
a basic building block for patterns in families of certain graphs. These relations lead to a
couple of sums of graphs: closed and open Zykov sums denoted +R , ⊕R respectively. Both
are defined below.
2
Definition 1. Let ⊎ be the disjoint union operator. Let G1 , . . . , Gn be a collection of
graphs and R1 , . . . , Rn−1 be a collection of relations such that each Ri is a relation set from
V (Gi ) to V (Gi+1 ), i = 1, . . . , n − 1 . Let Ei = {{v, w} ⊆ V (Gi ) ⊎ V (Gi+1 ) | v ∈ V (Gi ), w ∈
V (Gi+1 ) with vRi w}, i = 1, . . . , n − 1. We define a new graph G1 +R1 · · · +Rn−1 Gn as follows:
V (G1 +R1 · · · +Rn−1 Gn ) = V (G1 ) ⊎ · · · ⊎ V (Gn )
and
E(G1 +R1 · · · +Rn−1 Gn ) = E(G1 ) ⊎ · · · ⊎ E(Gn ) ⊎ E1 ⊎ · · · ⊎ En−1 .
We call G1 +R1 · · · +Rn−1 Gn an open Zykov sum .
If we have an additional relation Rn from V (Gn ) to V (G1 ), we define an extra graph
G1 ⊕R1 · · · ⊕Rn−1 Gn ⊕Rn G1 as
V (G1 ⊕R1 · · · ⊕Rn−1 Gn ⊕Rn G1 ) = V (G1 ) ⊎ · · · ⊎ V (Gn )
and
E(G1 ⊕R1 · · · ⊕Rn−1 Gn ⊕Rn G1 ) = E(G1 ) ⊎ · · · ⊎ E(Gn ) ⊎ E1 ⊎ · · · ⊎ En
where En = {{u, v} : u ∈ V (Gn ) and v ∈ V (G1 ) with uRn v}. We call G1 ⊕R1 · · · ⊕Rn−1
Gn ⊕Rn G1 a closed Zykov sum .
In the following section, we present several well known families of graphs that can be
generated from Zykov sums.
2.1
Examples
Let Cn be the n-cycle graph with V (Cn ) = {0, 1, . . . , n − 1} and E(Cn ) = {{0, 1}, . . . , {n −
2, n − 1}, {n − 1, 0}}, where n = 1, 2, . . .
Platonic graphs. The platonic graphs are made of the vertices and edges of the five
platonic solids. They can be constructed from open Zykov sums as is shown in Table 1.
Note that the relations used are sometimes functions, and if not, the inverse relation is a
function, except in the icosahedron case, where neither R1±1 nor R2±1 is a function.
Hypercubes. Let Q0 be the graph with vertex set given by the singleton set {v} and empty
set of edges. Then the hypercubes Qn can be defined recursively by Qn+1 = Qn +Idn Qn ,
n ≥ 0 where Idn is the identity map on V (Qn ).
Paths. Let P0 be the singleton graph, where V (P0 ) = {v0 } and E(P0 ) = ∅. Let Id :
{v0 } → {v0 } be the identity map. Then
Pn = P0 +Id P0 +Id · · · +Id P0
is the path graph of length n − 1, where P0 appears n times.
3
Graph name
Zykov sum
Relation
Tetrahedral
Octahedral
C1 +R C3
C3 +R C3
Cube
Icosahedral
C4 +Id C4
C 3 + R1 C 6 + R2 C 3
Dodecahedral
C5 +R1 C10 +R2−1 C5
R = {(0, 0), (0, 1), (0, 2)}
R = {(0, 0), (0, 2), (1, 1), (1, 0),
(2, 2), (2, 1)}
Id identity map on V (C4 )
R1 = {(0, 0), (0, 1), (0, 2), (1, 2),
(1, 3), (1, 4), (2, 4), (2, 5), (2, 0)}
R2 = {(0, 0), (1, 0), (1, 1), (2, 1),
(3, 1), (3, 2), (4, 2), (5, 2), (5, 0)}
R1 , R2 : V (C5 ) → V (C10 )
R1 (i) = 2i, R2 (i) = 2i + 1
Table 1: The platonic graphs as open Zykov sums. The graphs Cn are the n-cycle graphs
Grids, cylinders and tori. The grid Gn,m is defined by V (Gn,m ) = {(i, j) : 1 ≤ i ≤
n, 1 ≤ j ≤ m} and E(Gn,m ) = {{(i, j), (u, v)} : |i − u| + |j − v| = 1}. Then
Gn,m = Pn +Id Pn +Id · · · +Id Pn
(1)
where the path Pn appears m times and Id is the identity function on V (Pn ).
Now, let Cn be the n-cycle graph and Id the identity map on V (Cn ). The cylinder n × m,
as an open Zykov sum, is
Cn,m = Cn +Id Cn +Id · · · +Id Cn
where Cn appears m times. On the other hand, the closed Zykov sum
Tn,m = Cn ⊕Id Cn ⊕Id · · · ⊕Id Cn
is the torus n × m, where Cn appears m + 1 times. Note that in the Zykov sum Tn,m , the
first and last terms have been identified by definition (see Definition 1).
3
Counting independent sets
In order to count independent sets, we are using the transfer matrix method [2, 6, 7, 10],
which is based upon a perpendicularity concept. Following this idea, we propose a new inner
product defined with the help of the relation set given in a Zykov sum.
Definition
Let B V (G) be the cartesian
Q 2. Let B = {0, 1} and let G, H be a pair of graphs.
V (H)
product v∈V (G) Bv , where each Bv = B; similarly for B
. Let R be a relation from
V (G)
V (H)
V (G) to V (H). For any a ∈ B
,b ∈ B
we define
X
ha|biR =
πv (a)πw (b) ∈ N
v,w
vRw
where πv : B V (G) → Bv = B, πw : B V (H) → Bw = B are the canonical projections.
4
Let 2V (G) be the power set of V (G). There exists a unique bijection ΨG : 2V (G) → B V (G)
such that
πv ΨG (A) = χA (v), ∀v ∈ V (G), ∀A ⊂ V (G)
(2)
where χA is the characteristic function of the set A.
The following lemma translates the concept of independent set in Zykov sums into perpendicularity.
Lemma 3. Let G, H be a pair of graphs and R be a relation from V (G) to V (H). If A, B
are independent sets of G, H respectively, then A ⊎ B is an independent set of G +R H if and
only if hΨG (A)|ΨH (B)iR = 0.
Proof. The disjoint union A ⊎ B is an independent set of G +R H iff for any v ∈ V (G), w ∈
V (H), vRw implies {v, w} 6⊂ A ⊎ B iff vRw implies χA (v)χB (w) = 0 iff
X
πv (ΨG (A))πw (ΨH (B)) = 0
v,w
vRw
because of (2).
The transfer matrix is defined below using the inner product relative to the Zykov sums.
Definition 4.
1. For a graph G, we denote the collection of independent sets of G with
IG ; while F (G), called the Fibonacci number of G, stands for the cardinality of IG ,
i.e., F (G) = |IG |.
2. For any z ∈ N we define
z=
(
1,
0,
if z = 0;
otherwise.
3. Let G, H be a pair of graphs and R be a relation from V (G) to V (H). The function
TR
G,H : IG × IH → B,
TR
G,H (A, B) = hΨG (A)|ΨH (B)iR
is called the transfer matrix of G +R H.
Note that for any z1 , z2 ∈ N, z1 + z2 = z1 z2 . In the following Theorems 5 and 6, we show
how to calculate the Fibonacci number of open and closed Zykov sums.
Theorem 5. Let G, H be a pair of graphs and R be a relation from V (G) to V (H). Then
1.
X
F (G +R H) =
A∈IG ,B∈IH
where TR
G,H is the transfer matrix of G +R H.
5
TR
G,H (A, B)
2. If G = H,
F (G ⊕R G) =
X
TR
G,G (A, A)
A∈IG
where TR
G,G is the transfer matrix of G +R G.
Proof.
1. We have that if J ∈ IG+R H then there exist A ∈ IG and B ∈ IH such that
J = A ⊎ B. Now, use Lemma 3.
2. We have that J ∈ IG⊕R G iff J ∈ IG and hΨ(J)|Ψ(J)iR = 0.
Theorem 6. Let G, H, K be graphs, R be a relation from V (G) to V (H) and S be a relation
from V (H) to V (K). Then,
X
X
S
F (G +R H +S K) =
TR
(3)
G,H (A, B)TH,K (B, C).
A∈IG ,C∈IK B∈IH
Furthermore, if K = G, then
X
F (G ⊕R H ⊕S G) =
S
TR
G,H (A, B)TH,G (B, A) .
(4)
A∈IG ,B∈IH
Proof. We have that J ∈ IG+R H+S K iff there exist A ∈ IG , B ∈ IH , C ∈ IK such that
J = A ⊎ B ⊎ C and, due to Lemma 3,
hΨG (A)|ΨH (B)iR + hΨH (B)|ΨK (C)iS = 0
which is equivalent to
hΨG (A)|ΨH (B)iR
so
|IG+R H+S K | =
X
hΨH (B)|ΨK (C)iS = 1
hΨG (A)|ΨH (B)iR
hΨH (B)|ΨK (C)iS
A∈IG ,B∈IH
C∈IK
from which (3) follows.
Similarly J ∈ IG⊕R H⊕S G iff there exist A ∈ IG and B ∈ IH such that J = A ⊎ B and
hΨG (A)|ΨH (B)iR + hΨH (B)|ΨG (A)iS = 0, since Lemma 3. Thus
hΨG (A)|ΨH (B)iR
hΨH (B)|ΨG (A)iS = 1
which leads to
|IG⊕R H⊕S G | =
X
hΨG (A)|ΨH (B)iR
A∈IG ,B∈IH
6
hΨH (B)|ΨG (A)iS .
In fact, the transfer matrix is an actual matrix indexed by the cartesian product of
independent sets IG × IH . We denote such matrix by
R
TR
G,H = (TG,H (A, B))A∈IG ,B∈IH .
Note that TR
G,H depends on the inner product which is denoted as a bracket. This notation
is borrowed from the Dirac notation used mainly in quantum mechanics, where it has proved
of great value. So, we are going to keep using this notation when we define, for any graph
G, the column matrix full of 1’s indexed by the vertices of G which we denoted |Gi, while
hG| is the transpose of |Gi. Then, Theorem 6 ensures that
S
F (G + H + K) = hG|TR
G,H TH,K |Ki,
(5)
S
t
F (G ⊕R H ⊕S G) = hG|TR
G,H ∗ (TH,G ) |Hi
(6)
furthermore
where ∗ stands for the Hadamard matrix product and the superindex t indicates matrix
transposition. Let us recall that the Hadamard matrix product A ∗ B of two matrices
A = (ai,j ), B = (bi,j ) of the same dimensions is given by multiplying the corresponding
entries together: A ∗ B = (ai,j bi,j ). Also, note that we can write the right hand side of (6)
as a standard matrix product as follows
S
F (G ⊕R H ⊕S G) = Tr TR
(7)
G,H TH,G
where Tr denotes the matrix trace. However the formula in (6) is less difficult to calculate
than (7), since the fastest known algorithms for computing the usual matrix product have
complexity strictly greater than quadratic, which is the complexity of the Hadamard matrix
product. Thus, the formula in (6) is useful for computer calculation.
A similar formula to (5) and (6), for a general graph, was found by Merrifield and Simmons
[12, p. 209] using an exponential operator related to the annihilation and creation operators,
instead of our transfer matrices and bra and ket vectors. However, said Merrifield-Simmons
formula is not a convenient way to handle the Zykov sum structure.
With the aforementioned notation, the proof of Theorem 6 can be generalized in order
to obtain:
Theorem 7.
1. The number of independent sets of G1 +R1 G2 +R2 · · · +Rn−1 Gn is the
Rn−1
1
matrix product hG1 |TR
G1 ,G2 · · · TGn−1 ,Gn |Gn i.
2. The number of independent sets of G1 ⊕R1 G2 ⊕R2 · · · ⊕Rn−1 Gn ⊕Rn G1 is the matrix
Rn−1
Rn
1
trace Tr(TR
G1 ,G2 · · · TGn−1 ,Gn TGn ,G1 ).
Next, the Fibonacci numbers of two classical platonic solid graphs are calculated using
Theorem 7. The remaining classical platonic solid cases are similar. Our goal here is to
show that the concept of Zykov sum is an adequate framework for using the matrix transfer
method for counting independent sets.
Throughout this paper, we calculate the transfer matrices after the lexicographic order
in the cartesian product {0, 1}|V (G)| induced by 0 < 1, for any given graph G.
7
Octahedral graph. Using the decomposition
given in Table 1, we get the following matrices:

1

1
hC3 | = (1, 1, 1, 1), TR
C3 ,C3 = 
1
1
of the octahedral graph as the Zykov sum
1
0
1
0

1
1
,
0
0
1
0
0
1
 
1
 1

|C3 i = 
 1 .
1
Thus, from Theorem 7, the Fibonacci number of the octahedral graph is F (C3 +R C3 ) =
hC3 |TR
C3 ,C3 |C3 i = 10.
Dodecahedral graph. From Table 1, we get that the dodecahedral graph G has Fibonacci
t
R2
1
|C5 i = 5, 828, where
number F (G) = hC5 |TR
C5 ,C10 TC5 ,C10

1
TR
C5 ,C10
4







t 
R2
TC5 ,C10 = 








123
89
89
89
65
89
65
65
89
65
65
89
55
55
63
39
64
40
40
63
39
45
89
63
55
55
39
63
45
39
64
40
40
89
64
63
55
40
55
40
39
63
45
39
65
40
39
39
24
40
25
24
45
27
27
89
63
64
63
45
55
39
40
55
40
39
65
39
40
45
27
40
24
25
39
24
27
65
45
40
39
27
39
27
24
40
25
24
89
55
63
64
40
63
39
45
55
39
40
65
39
39
40
24
45
27
27
40
24
25
65
40
45
40
25
39
24
27
39
27
24









 .








The series of independent sets
In this section, we study the generating function of sequences of Fibonacci numbers for
families of graphs determined by Zykov sums which are structures defined by repeating a
fixed pattern of relations.
In the following, we are assuming that G is a family (Gi )i∈N of graphs.
Definition 8. We call the infinite series
FG (x) = F (G0 ) + F (G1 )x + F (G2 )x2 + · · ·
(8)
the Fibonacci series of G.
The following families, which contain repetitive patterns of open and closed Zykov sums,
include many interesting cases such as grids, tori, cylinders [2, 7, 10] and so on.
Definition 9. Let G = (Gn )n≥0 be a family of graphs. We call G a family of:
1. Strip graphs if there exists a graph G such that G0 = G, G1 = G +R G, G2 =
G +R G +R G, . . . .
8
2. Ring graphs if there exists a graph G such that G0 = G, G1 = G ⊕S G, G2 = G ⊕R
G ⊕S G, G3 = G ⊕R G ⊕R G ⊕S G, . . . . In such a case G +S G is called the skewing of
G.
3. Alternating strip graphs if there exists a graph G such that G0 = G, G1 = G +R G,
G2 = G +R +G +R−1 G, G3 = G +R G +R−1 G +R G, . . . .
4. Alternating ring graphs if there exists a graph G such that G0 = G, G1 = G ⊕R G,
G2 = G ⊕R G ⊕R−1 G, G3 = G ⊕R G ⊕R−1 G ⊕R G, . . . .
In any case G is called the shape of G and G +R G is called the fundamental pattern of G.
Note that in the ring cases, the fundamental pattern is an open Zykov sum, while the
family elements are closed Zykov sums.
In the following theorems, we show that the families given in Definition 9 have Fibonacci
series with minor variations of the geometric series.
Theorem 10. Let G be a family of graphs as in Definition 9. Let G be the shape of G, T
the transfer matrix of the fundamental pattern and I the identity matrix.
1. If G is a family of strip graphs then
FG (x) = hG|(I − xT)−1 |Gi.
(9)
2. If G a family of ring graphs then
FG (x) = F (G) + Tr x(I − xT)−1 T1
where T1 is the transfer matrix of the skewing of G.
(10)
Proof.
1. From Theorem 7, we get, for any n non-negative integer, F (Gn ) = hG|Tn |Gi. So
FG (x) = hG|
∞
X
Tn xn |Gi = hG|(I − xT)−1 |Gi.
n=0
2. Similarly, from Theorem 7, we have F (Gn ) = Tr Tn−1 T1 , n ≥ 1. Then
FG (x) = F (G) +
∞
X
n=1
Tr Tn−1 T1 xn
= F (G) + Tr x(I − xT)−1 T1
due to the linearity of the matrix trace.
Similarly, we can prove the following.
9
Theorem 11. Under the notation given in Theorem 10.
1. If G is a family of alternating strip graphs then
FG (x) = hG|(I − x2 TTt )−1 (I + xT)|Gi.
(11)
2. If G is a family of alternating ring graphs then
FG (x) = Tr (I − x2 TTt )−1 (I + xT) .
5
Examples
We choose some interesting examples in order to illustrate our methods. We deal with
several particular cases of strips, which include cylinders, grids, nanotubes and some kind of
cylinders with Petersen graph shape as well as their closed versions as rings: tori, generalized
Möbius strips, nanotori, and tori with Petersen graph shape.
Centipedes. Now, our fundamental pattern is P2 +f P2 given in Figure 1 where the partial
function f : {0, 1} → {0, 1} is defined just by f (0) = 0. Next, we take the family of strip
Figure 1: Fundamental pattern of the centipedes
graphs G = (Gn )n≥0 given by open Zykov sums Gn = P2 +f · · · +f P2 , where the number of
P2 is n + 1, n = 0, 1, . . . Then, the transfer matrix of P2 +f P2 is


1 1 1
(12)
TfP2 ,P2 = 1 0 1 .
1 1 1
Then, from (9), we get
3 + 2x
1 − 2 x − 2 x2
which is the generating function, except for the first term, of the sequence A028859 in [19].
FG (x) =
10
Figure 2: Strip of graphs with fundamental pattern P3 +R P3
Diagonal grids. Let P3 be the path graph with set of vertices {0, 1, 2}. The family G of
strip graphs given in Figure 2 has fundamental pattern P3 +R P3 where R is the relation on
V (P3 ) defined by 1R 0, 2R 1 (see Figure 3).
Its transfer matrix is


1 1 1 1 1
 1 1 0 1 1


R
.
1
1
1
0
0
TP3 ,P3 = 


 1 1 1 1 1
1 1 0 1 1
From (9) it follows that the corresponding Fibonacci series is
FG (x) = −
4x − 5
3 x2 − 5 x + 1
which is the generating function of the sequence A188707. Similarly, we have that the family
of graphs R given in Figure 4 is a family of ring graphs with fundamental pattern and
skewing P3 +R P3 . From (10) we have that its Fibonacci series is
FR (x) =
9 x2 − 20 x + 5
.
3 x2 − 5 x + 1
Figure 3: The fundamental pattern P3 +R P3
11
Figure 4: A generic element of the family R
Cylinders. Let Ci be the i-cycle graph and Pn be the path graph with n vertices. Let
Id : V (Ci ) → V (Ci ) be the identity map. Then, the cylinder Ci × Pn is G(i, n) = Ci +Id
Ci +Id · · · +Id Ci , where the number of cycles is n + 1, n ≥ 0. Thus, the cylinders G(i, ∗)
form a family of strip graphs with fundamental pattern Ci +Id Ci and shape Ci . From (9),
for some fixed i, we can calculate the generating function FG(i,∗) (x), as shown in Table 2.
i
FG(i,∗) (x)
Sequence
2
3
4
5
6
7
8
x+3
− x2 +2
x−1
x+4
− x2 +3x−1
2 −7
− x3 −xx2 −5x+1
2 −4x−11
− x3x−5x
2 −7x+1
A078057
A003688
A051926
A181989
A181961
A182014
A182019
4
7
6
3
2
x −x −25x −17x+18
− x5 −2x
4 −25x3 −3x2 +12x−1
4 +6x3 −38x2 −16x+29
− x5x+5x
4 −44x3 +8x2 +17x−1
5
4
3
2
+6x −105x +108x +394x −163x −208x+47
− x8x+5x
7 −109x6 +187x5 +334x4 −317x3 −65x2 +29x−1
Table 2: The Fibonacci series of some cylinders. These are the generating functions of the
integer sequences in the third column except for the first term
Tori. Let Ci be the cycle graph of length i. Then G(i, n) = Ci ⊕Id Ci ⊕Id · · · ⊕Id Ci ⊕Id Ci
where the number of cycles written is n + 1, n = 0, 1, . . .. Such graph is the torus Ci × Cn .
Thus, the family of ring graphs G(i, ∗) has fundamental pattern and skewing Ci +Id Ci .
Again, we can calculate FG(i,∗) for some particular values of i with the help of (10), as shown
in Table 3.
Generalized Petersen graphs. We are dealing with generalized Petersen graphs P (i, 2)
defined by V (P (i, 2)) = {0, 1, . . . , 2i − 1} and
E(P (i, 2)) = {j, j + i} | 0 ≤ j ≤ i − 1 ∪ {j, k} | j − k ≡ 0 (mod 2) and i ≤ j, k < 2i
∪ {j, k} | k ≡ j + 1 (mod i) and 0 ≤ j, k ≤ i − 1 .
12
i
FG(i,∗) (x)
Sequence
3
4
5
6
5 x2 +7 x−4
x3 +4 x2 +2 x−1
5 +17 x4 −20 x3 −72 x2 −13 x+7
− xx6 −2
x5 −7 x4 +8 x3 +15 x2 +2 x−1
11 x6 −27 x5 −130 x4 +70 x3 +220 x2 +43 x−11
− x7 −8 x6 +7 x5 +30 x4 −10 x3 −27 x2 −4 x+1
A051928
A050402
A182041
A182052
−p(x)/q(x)
Table 3: The Fibonacci series of several tori. In the case i = 6 we have p(x) = 9 x12 +67 x11 −
556 x10 −1162 x9 +6841 x8 −1421 x7 −12335 x6 +3985 x5 +7340 x4 −1182 x3 −1317 x2 −71 x+18
and q(x) = x13 − 4 x12 − 36 x11 + 119 x10 + 295 x9 − 1032 x8 + 115 x7 + 1301 x6 − 360 x5 −
575 x4 + 89 x3 + 84 x2 + 4 x − 1
The Fibonacci number for generalized Petersen graphs P (i, 2) with i an odd number was
calculated by Wagner [20]. Here we calculate the Fibonacci series for the family of generalized
Petersen graphs using the transfer matrix method. Let P4 be the the path graph with vertices
V (P4 ) = {0, 1, 2, 3}. Let R be the relation on V (P4 ) defined by 0R 0, 3R 3, 2R 1 (see Figure
5).
Figure 5: Fundamental pattern in the Petersen graphs P (2i, 2)
Then, the ring graph family with shape P4 , fundamental pattern P4 +R P4 and skewing
the same P4 +R P4 , is the family of generalized Petersen graphs G = P (2i, 2) i≥0 :
P (2i, 2) = P4 ⊕R P4 ⊕R · · · ⊕R P4 ⊕R P4 ,
where P4 appears i+1 times, i = 0, 1, . . .. Note that we included the non-standard generalized
Petersen graphs P (0, 2) = P4 and the multigraph P (2, 2) = P4 ⊕R P4 .
The related transfer matrix is


1 1 1 1 1 1 1 1
1 0 1 1 0 1 0 1


1 1 1 0 0 1 1 1


1 1 1 1 1 1 1 1
R

TP4 ,P4 = 
(13)
1 0 1 1 0 1 0 1 .


1 1 1 1 1 0 0 0


1 0 1 1 0 0 0 0
1 1 1 0 0 0 0 0
13
Then, from (10), we get
FG (x) =
(6 x2 − 11 x − 8) (2 x3 − 5 x2 − 4 x + 1)
4 x5 − 13 x4 + 3 x3 + 15 x2 + 3 x − 1
which is the generating function of A182054.
For generalized Petersen graphs of the type P (2i + 1, 2), i ≥ 2 we take the family P given
in Figure 7, i.e.,
Mi−1
P (2i + 1, 2) =
P4 ⊕R2 M ⊕R3 P4
R1 k=1
where M is the graph
such
that
V
(M
)
=
{0,
1,
2,
3,
4,
5},
E(M
)
=
{0, 1}, {1, 2}, {2, 3},
{2, 4}, {4, 5}, {0, 5} and relations R1 = (0, 0), (3, 3), (2, 1) , R2 = (0, 0), (3, 3), (2, 1) ,
R3 = {(3, 0), (5, 3), (4, 1)}.
From the Theorem 7 and proof of (10) in Theorem 10, we get
FP (x) =
∞
X
Tr(Tj1 T2 T3 )xj = Tr (I − xT1 )−1 T2 T3
j=0
R2
R3
1
where T1 = TR
P4 ,P4 is given by (13); while T2 = TP4 ,M and T3 = TM,P4 are 8 × 19 and 19 × 8
matrices respectively. Thus,
FP (x) = −
52 x4 − 165 x3 + 16 x2 + 207 x + 76
4 x5 − 13 x4 + 3 x3 + 15 x2 + 3 x − 1
which is the generating function of A182077 with a shift of one term.
Families with Petersen graph shape. Let Id : V (P (5, 2)) → V (P (5, 2)) be the identity
map. We take G as the family of strip graphs with fundamental pattern P (5, 2) +Id P (5, 2)
and shape of the Petersen graph P (5, 2), i.e., G = (Gn )n≥0 where Gn = P (5, 2)+Id P (5, 2)+Id
· · · +Id P (5, 2) with n + 1 copies of the Petersen graph, n ≥ 0 (see Figure 8). Then, the
transfer matrix TId
P (5,2),P (5,2) is a 76 × 76-matrix, since F (P (5, 2)) = 76.
Figure 6: Generalized Petersen graph P (2i, 2).
14
Figure 7: The family of graphs P (2i + 1, 2)
From (9) we get
FG (x) = −
x5 + 12 x4 − 130 x3 + 92 x2 + 237 x + 76
.
x6 + 11 x5 − 137 x4 + 172 x3 + 215 x2 + 39 x − 1
Similarly, for G ′ the family of ring graphs with fundamental pattern and skewing given by
P (5, 2) +Id P (5, 2), we get from (10), that its Fibonacci series FG ′ (x) satisfies FG ′ (x) =
p(x)/q(x) where
p(x) = 59 x20 + 158 x19 − 17410 x18 − 31425 x17 + 843564 x16 + 1040034 x15
− 10876134 x14 − 9246646 x13 + 52315426 x12 + 29197770 x11 − 101636518 x10
− 28773932 x9 + 77606056 x8 + 9105678 x7 − 21502410 x6 − 847682 x5 + 1979331 x4
+ 80616 x3 − 50408 x2 − 2203 x + 76
and
q(x) = (x − 1) x2 + 2 x − 1
x5 + 5 x4 − 4 x3 − 14 x2 + 3 x + 1
x6 + 11 x5 − 137 x4 + 172 x3 + 215 x2 + 39 x − 1
x7 + 4 x6 − 52 x5 − 105 x4 + 51 x3 + 78 x2 − 10 x − 1 .
We obtain an additional family G ′′ if, instead of closing with the relation given by the
identity map, we take a rotation R by an angle of 2π/5. More formally, we take V (P (5, 2)) =
Figure 8: A strip graph with shape P (5, 2), the Petersen graph. The edges given by the
identity map are shown by dotted lines
15
Figure 9: A pair of rings with shape P (5, 2), the Petersen graph. The edges given
by the relation maps are shown by dotted lines. The ring on the left has relation
maps the identity map; while the ring on the right has skewing given by the relation R = {(0, 1) , (1, 2) , (2, 3) , (3, 4) , (4, 0) , (5, 6) , (6, 7) , (7, 8) , (8, 9) , (9, 5)}. The former
ring has Fibonacci number 27,053,615,385,404,201. The latter has Fibonacci number
27,050,814,022,108,001.
{0, 1, 2, . . . , 9} as the vertices set of the Petersen graph, and
E(P (5, 2)) = {{0, 1} , {0, 4} , {0, 5} , {1, 2} , {1, 6} , {2, 3} , {2, 7} , {3, 4} ,
{3, 8} , {4, 9} , {5, 7} , {5, 8} , {6, 8} , {6, 9} , {7, 9}} .
Then, the new skewing induced by R is defined as follows (see Figure 9)
R = {(0, 1) , (1, 2) , (2, 3) , (3, 4) , (4, 0) , (5, 6) , (6, 7) , (7, 8) , (8, 9) , (9, 5)} .
Then FG ′′ (x) = p1 (x)/q1 (x), where
p1 (x) = 75 x15 + 1284 x14 − 8828 x13 − 101662 x12 + 376556 x11 + 1642004 x10
− 2174799 x9 − 7893320 x8 + 252699 x7 + 6559072 x6 + 1031350 x5 − 1259454 x4
− 160398 x3 + 47456 x2 + 2305 x − 76
and
q1 (x) = x2 + 2 x − 1
x6 + 11 x5 − 137 x4 + 172 x3 + 215 x2 + 39 x − 1
x7 + 4 x6 − 52 x5 − 105 x4 + 51 x3 + 78 x2 − 10 x − 1 .
Armchair nanotube graphs. Let Bn be the graph with 2n vertices {0, 1, . . . , 2n − 1}
and set of edges {{0, 1}, {2, 3}, . . . , {2n − 2, 2n − 1}}, i.e., Bn is the disjoint union of n copies
of the one-length path P2 :
Bn = P2 +∅ · · · +∅ P2 .
We define a relation map R : V (Bn ) → V (Bn ) as R(i) = (i − 1) mod 2n, i = 0, . . . , 2n − 1.
An armchair nanotube graph of length n and breadth k is
N Tn,k = Bn +R Bn +R−1 Bn +R · · · +R±1 Bn
16
where Bn appears k + 1 times, k ≥ 0 (this graph forms the structure of the (n, n) armchair
carbon nanotube [14] without caps). Thomassen [16] defines a similar graph called hexagonal
cylinder circuit, however this belongs to a different family of nanotubes: zig-zag carbon
nanotubes.
By definition N Tn,∗ = N Tn,k k≥0 is a family of alternating strip graphs with fundamental
pattern Bn +R Bn . By (11) and a calculation similar to those in the previous examples, we
get that the Fibonacci series of the armchair nanotube graphs of length 3 is
FN T3,∗ (x) = −
28 x4 + 55 x3 − 89 x2 − 29 x + 27
28 x5 + 42 x4 − 109 x3 + 17 x2 + 13 x − 1
which is the generating function of A182130.
Similarly, a nanotorus graph is the following closed Zykov sum:
N τn,k = Bn ⊕R Bn ⊕R−1 Bn +R · · · ⊕R±1 Bn ,
where, again, Bn appears k + 1 times, k ≥ 0. Now the family N τ3,∗ = (N τ3,k )k≥0 has
Fibonacci series FN τ3,∗ (x) = p(x)/q(x), where
p(x) = 979776 x18 − 75600 x17 − 12197940 x16 + 5916552 x15 + 35833019 x14
− 19220271 x13 − 44070216 x12 + 23310438 x11 + 26177559 x10 − 13274349 x9
− 7520073 x8 + 3654387 x7 + 940365 x6 − 451464 x5 − 43362 x4 + 24495 x3
+ 25 x2 − 468 x + 27
and
q(x) = (x − 1) (x + 1) 3 x3 − 5 x2 − 5 x + 1 36 x4 − x3 − 20 x2 − x + 1
36 x4 + x3 − 20 x2 + x + 1 28 x5 + 42 x4 − 109 x3 + 17 x2 + 13 x − 1 .
The rational function FN τ3,∗ (x) is the generating function of the sequence A182141.
Generalized Möbius Ladders. Let Pn be the path graph with n vertices, V (Pn ) =
{0, . . . , n − 1}, σ be a permutation of V (Pn ) and Id be the identity map on V (Pn ). Then a
generalized Möbius ladder M (n, m, σ) is
M (n, m, σ) = Pn ⊕Id · · · ⊕Id Pn ⊕σ Pn
where Pn appears m + 1 times, m = 0, 1, . . .; for instance, if σ : V (P2 ) → V (P2 ) is the
transposition σ(0) = 1 and σ(1) = 0 then M (2, ∗, σ) = (M (2, m, σ))m≥0 is the family of usual
Möbius ladders. The family of ring graphs M (2, ∗, σ) has fundamental pattern P2 +Id P2 and
skewing P2 +σ P2 . We have the transfer matrices,




1 1 1
1 1 1
1 0 1 , TσP ,P = 1 1 0 .
TId
P2 ,P2 =
2 2
1 0 1
1 1 0
17
Figure 10: The nanotube graph N T5,21 on the left and the nanotorus graph N τ5,21
on the right. Once again, the edges given by the relation set are shown by dotted
lines. Using Theorem 7 we get that the nanotube graph N T5,21 has Fibonacci number
14,890,453,762,710,452,477,470,450,680,772,895,445,343; while the nanotorus graph N τ5,21
has Fibonacci number 73,562,247,493,061,556,896,479,759,292,362,159,745
Then, from (10), we get
FM (2,∗,σ) (x) =
2 x3 + 7 x2 − 3
(x + 1) (x2 + 2 x − 1)
which is the generating function of A182143 except for the first term.
Now, we take families M (3, ∗, σi ) = (M (3, m, σi ))m≥0 , i = 1, 2, 3 with σ1 , σ2 , σ3 permutations of V (P3 ) defined in Figure 11. Thomassen [16] calls the family of graphs M (3, σ2 )
quadrilateral Möbius double circuits.
We have




1 1 1 1 1
1 1 1 1 1
 1 1 1 0 0
 1 0 1 1 0




σ2
σ1



TP3 ,P3 = 1 1 1 0 0 , TP3 ,P3 = 
 1 1 0 1 1 ,
 1 0 1 1 0
 1 1 0 1 1
1 0 1 0 0
1 0 0 1 0




1 1 1 1 1
1 1 1 1 1
 1 0 1 1 0
 1 1 0 1 1




σ3
Id
 , TP ,P = 1 1 0 1 1 .
1
1
1
0
0
TP3 ,P3 = 
3 3




 1 1 1 0 0
 1 0 1 1 0
1 0 1 0 0
1 0 0 1 0
Then
FM (3,∗,σ1 ) (x) =
3 x5 + 6 x4 − 19 x3 − 30 x2 − 2 x + 5
,
(x + 1) (x4 − 6 x2 − 2 x + 1)
2 x5 + x4 − 24 x3 − 28 x2 − 2 x + 5
FM (3,∗,σ2 ) (x) =
,
(x + 1) (x4 − 6 x2 − 2 x + 1)
and
4 x5 + 6 x4 − 25 x3 − 32 x2 − x + 5
.
FM (3,∗,σ3 ) (x) =
(x + 1) (x4 − 6 x2 − 2 x + 1)
18
Figure 11: A subset of permutations of V (P3 )
Some regular and almost regular graphs. Let Bn be the disjoint union of n copies of
the singleton graph K1 , i.e, V (Bn ) = {0, . . . , n − 1} and E(Bn ) = ∅. Let R be the relation
on V (Bn ) given by iRi and iRj where j ≡ i − 1 (mod n), i = 0, . . . n − 1. Note that R is
also a relation from vertices of the n-cycle Cn to those of Bn . We define
Gkn = Cn +R Bn +R · · · +R Bn
Rnk+1 = Cn +R Bn +R · · · +R Bn +R Cn
where Bn appears k − 1 times, k = 1, 2, . . . and G0n = Rn0 = ∅ the empty graph, Rn1 = Cn .
Furthermore, let Cn′ be the complement graph of the n-cycle graph Cn and Kn the complete
graph on n vertices with V (Kn ) = {0, 1, . . . , n − 1}. Also, we define
Knk = Cn′ +R · · · +R Cn′ +R Kn
Pnk+1 = Kn +R Cn′ +R · · · +R Cn′ +R Kn
where Cn′ appears k − 1 times, k = 1, 2, . . . and Kn0 = Pn0 = ∅, Pn1 = Kn . The graphs
Gkn , Rnk , Knk and Pnk are called (almost) regular graphs class 1, class 2, class 3 and class 4,
respectively by Burstein, Kitaev, and Mansour [1]. Thus, from Theorem 7, we get
−1
2
R
FG∗n (x) = 1 + F (Cn ) x + hCn |TR
Cn ,Bn (I − x TBn ,Bn ) |Bn i x ,
−1 R
R
3 R
)
T
(I
−
x
T
+
x
T
FRn∗ (x) = 1 + F (Cn ) x + hCn | x2 TR
Bn ,Cn |Cn i,
Bn ,Bn
Cn ,Bn
Cn ,Cn
−1 R
FKn∗ (x) = 1 + (n + 1) x + hCn′ |(I − x TR
TCn′ ,Kn |Kn i x2 ,
′ ,C ′ )
Cn
n
(14)
(15)
(16)
3 R
R
−1 R
|Kn i. (17)
FPn∗ (x) = 1 + (n + 1) x + hKn | x2 TR
T
′ (I − x TC ′ ,C ′ )
′ ,K
C
Kn ,Kn + x TKn ,Cn
n
n n
n
Burstein et al [1] introduce algorithms for computing the Fibonacci series of the families
G∗n , Rn∗ , Kn∗ , and Pn∗ . However these algorithms fail for the families Rn∗ , Kn∗ and Pn∗ . For
instance, it is easy to see that F (R33 ) = 32, while 32 does not appear in the sequence
A026150 which is the Fibonacci numbers sequence of the family R3∗ , according to Burstein
et al. Also F (K42 ) = 25, F (P43 ) = 77 are counterexamples to Theorem 3.3 and Theorem 3.4
of [1], respectively. As a consequence, since 77 does not appear in A007483, Burstein et al
19
n FRn∗ (x)
3
4
5
6
2
− 2 x42x−4−1
x+1
10 x5 +41 x4 −38 x3 −14 x2 +4 x+1
7 x4 +15 x3 −14 x2 −3 x+1
55 x6 −193 x5 −303 x4 +149 x3 +39 x2 −6 x−1
22 x5 −31 x4 −69 x3 +30 x2 +5 x−1
516 x8 −248 x7 −3688 x6 +1834 x5 +2518 x4 −588 x3 −112 x2 +10 x+1
108 x7 −84 x6 −532 x5 +178 x4 +280 x3 −66 x2 −8 x+1
FKn∗ (x)
Sequence
1
2 x2 −4 x+1
3 x2 +2 x+1
x3 −7 x2 −3 x+1
4 x2 +x+1
x3 −7 x2 −5 x+1
5 x2 +1
x3 −7 x2 −7 x+1
A007070
Table 4: Some Fibonacci series of classes 2 and 3 graphs
[1] are wrongfully relating this sequence to the class 4 graphs. The correct Fibonacci series,
after the formulas (15), (16) and (17) are shown in Tables 4 and 5.
Due to Theorem 10, the family ∅, C3 , C3 +R C3 , C3 +R C3 +R C3 , . . . has Fibonacci series
(2 x + 1)/(2 x2 + 2 x − 1) which is the generating function of A026150; while the family
∅, K4 , K4 +R K4 , K4 +R K4 +R K4 , . . . has Fibonacci series −(2 x + 1)/(2 x2 + 3 x − 1) which
is the generating function of A007483.
n
FPn∗ (x)
Sequence
3
4
5
6
(2 x+1)
− (22x−1)
x2 −4 x+1
3 +5 x2 −2 x−1
− 8xx3 −7
x2 −3 x+1
10 x3 +11 x2 −x−1
− x3 −7 x2 −5 x+1
x3 +19 x2 −1
− x123 −7
x2 −7 x+1
A161941
Table 5: Some Fibonacci series of the class 4 graphs. The series FP3∗ (x) is the generating
function of two times the terms in A161941 except the first one
6
Acknowledgements
The authors would like to thank Luke Goodman for helpful comments and suggestions.
References
[1] A. Burstein, S. Kitaev, and T. Mansour, Counting independent sets in some classes of
(almost) regular graphs, Pure Math. Appl. (PU.M.A.) 19 (2008), 17–26.
[2] N. Calkin and H. S. Wilf, The number of independent sets in a grid graph, SIAM J.
Discrete Math. 11 (1997), 54–60.
[3] T. Carroll, D. Galvin, and P. Tetali, Matchings and independent sets of a fixed size in
regular graphs, J. Combin. Theory Ser. A 116 (2009), 1219–1227.
20
[4] J. Cutler and A. J. Radcliffe, Extremal graphs for homomorphisms, J. Graph Theory
67 (2011), 261–284.
[5] R. Diestel, Graph Theory, Springer, 2006.
[6] K. Engel, On the Fibonacci number of an m × n lattice, Fibonacci Quart. 28 (1990),
72–78.
[7] R. Euler, The Fibonacci number of a grid graph and a new class of integer sequences,
J. Integer Seq. 8 (2005), Article 05.2.6.
[8] D. Galvin, Two problems on independent sets in graphs, Discrete Math. 311 (2011),
2105–2112.
[9] D. Galvin and Y. Zhao, The number of independent sets in a graph with small maximum
degree, Graphs Combin. 27, (2011) 177–186.
[10] M. Golin, Y. C. Leung, Y. J. Wang, and X. R. Yong, Counting structures in grid-graphs,
cylinders and tori using transfer matrices: Survey and new results, in C. Demetrescu,
R. Sedgewick, and R. Tamassia, eds., Proceedings of the Second Workshop on Analytic
Algorithmics and Combinatorics (ANALCO05), SIAM, 2005, pp. 250–258.
[11] A. Knopfmacher, R. F. Tichy, S. Wagner, and V. Ziegler, Graphs, partitions and Fibonacci numbers, Discrete Appl. Math. 155 (2007), 1175–1187.
[12] R. E. Merrifield and H. E. Simmons, Topological Methods in Chemistry, Wiley, 1989.
[13] A. S. Pedersen and P. D. Vestergaard, The number of independent sets in unicyclic
graphs, Discrete Appl. Math. 152 (2005), 246–256.
[14] C. P. Poole, Jr. and F. J. Owens, Introduction to nanotechnology, Wiley, 2003.
[15] H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982),
16–21.
[16] C. Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive graphs
on a fixed surface, Trans. Amer. Math. Soc. 323 (1991), 605–635.
[17] A. A. Sapozhenko, The number of independent sets in graphs, Moscow Univ. Math.
Bull. 62 (2007), 116–118.
[18] I. Stojmenovic, Honeycomb networks: Topological properties and communication algorithms, IEEE Trans. Parallel Distrib. Syst. 8 (1997), 1036–1042.
[19] The On-Line Encyclopedia of Integer Sequences, http://oeis.org.
[20] S. G. Wagner, The Fibonacci number of generalized Petersen graphs, Fibonacci Quart.
44 (2006), 362–367.
21
[21] Y. Zhao, The number of independent sets in a regular graph, Combin. Probab. Comput.
19 (2010), 315–319.
2010 Mathematics Subject Classification: Primary 11B39; Secondary 05C76, 11B83, 05A15,
05C69.
Keywords: Fibonacci number, Merrifield-Simmons index, independent set, Zykov sum,
transfer matrix method.
(Concerned with sequences A003688 A007070 A007483 A026150 A028859 A050402 A051926
A051928 A078057 A161941 A181961 A181989 A182014 A182019 A182041 A182052 A182054
A182077 A182130 A182141 A182143 and A188707.)
Received May 4 2012; revised August 22 2012; September 13 2012. Published in Journal of
Integer Sequences, September 23 2012.
Return to Journal of Integer Sequences home page.
22